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1 Suggested Problems for of the Textbook 3rd Edition Suggested Problems #1: (Not to be handed in) 1.2 5,12( parameter), 23,26,27,28( interval) # ans: ans: 3.1 #25 3.2 #17,29 3.3 #21,22( 0.91),23,24( 1.095) ans: ans: Ch. 3 Review #1(omit d),9 16. Consider the Databank data set, described on the 'R Related Things' page on childsmath. Which variables are measured on the ordinal scale? ( education level, smoking status, ans: exercise) Suggested Problems #2: (Not to be handed in) Note: D o use the binomial or Poisson tables because they will be provided on a test. not not 4.1 #23,27,28( 0.125) ans: 4.2 #17,19,23,26( .995; .000260, 0.999740; .00000026, 0.99999974),29 ans: 4.3 #7,13,19 Ch. 4 Review #5,7,11 5.1 #4( no, .136),7,9,13,15,17 ans: 5.2 #13,16( ,9428),22( .2364),27,29,31,33,36( .101 to 7.899 no; .0695; .0861; c, no; ans: ans: ans: there is not strong evidence that the vaccine works) 5.3 #7,10( 0.497; 0.348; 0.122; 0.0284; 0.00497) ans: 31. Subjects for the California Health Interview Survey are contacted using telephone numbers in which the last four digits are randomly selected (with replacement). Find the probability that for one such phone number, the last four digits include at least one 0. ( .3439) ans: 32. Refer to the figures below showing surge protectors and used to protect an expensive : ; magnetic resonance imaging (MRI) scanner used in a hospital. If there is a surge in the voltage, the surge protector reduces it to a safe level. Assume that each surge protector has a .985 probability of working correctly when a voltage surge occurs. (a) If the two surge protectors are arranged in series, what is the probability that a voltage surge will not damage the MRI? (b) If the two surge protectors are arranged in parallel, what is the probability that a voltage surge will not damage the MRI? (ans: .999775; .970225)

2 33. autosomal recessive Suppose that a disease is inherited via an mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of of inheriting the disease. In a family of three children, find the probability that " % (a) All three children inherit the disease. (b) Only the first child inherits the disease. (c) Exactly one of the children inherits the disease. ( .015625, .140625, .421875) ans: 34. Three people are selected at random from a group of 20 men and 25 women, to participate in an exercise study. Find the probability that (a) All three of them are women. (b) Exactly two of them are men. (c) At least two of them are women. ( ) ans: #$! %(& )$! "%"* "%"* "%"* ß ß 35. Consider the R Output below which is a cross-tabulation table of the variables exercise ( and ( 0 1 2 3 0 1 = none, = light, = moderate, = heavy) = does not smoke, smoking status = smokes less than one pack per day, = smokes one or more packs per day 2 ) from the Databank data set. If two people are selected at random from the group of people summarized in the below output, find the probability that (a) Neither of them exercise. (b) Exactly one of them does not smoke. ( ) ans: (!$ #%*" %*&! %*&! ß smoking.status exercise 0 1 2 All 0 17 14 7 38 1 15 18 5 38 2 8 3 2 13 3 7 2 2 11 All 47 37 16 100 Suggested Problems #3: (Not to be handed in) 6.1 #11,37,39,41,43 6.2 #17,21,23,25,31 6.4 #5,11,17 6.5 #5,7,17 Ch. 6 Review #6( 131; .0001; no) ans: Ch. 6 Cumulative Review #3 7.1 #18( .691 to .785; no because the CI contains .75),19,35 ans: 7.2 #7,15,17,21 Ch. 7 Review #6

3 Suggested Problems #4: (Not to be handed in) 8.1 #1,5,7,13,15,17,19,21,23,25,27 8.2 #13,19,24( -value , fail to reject ),31 ans: D œ Þ''ß : œ Þ&!*# L ! 8.3 #15,18 critical value 1.677; fail to reject ),21,23 Ð > œ Þ"$$ß :  Þ"!ß L ans: ! 9.1 #7(a),(c),11(a),(c),15 9.2 #7(a),9,14(a),(c)( c.v. , Reject , yes),25 ans: > œ #Þ#)#ß :  Þ!&ß œ "Þ(#& L ! 12.1 #5,11 1. Consider the data set that is summarized in the R output. Find the missing -value. See the : following example for an explanation of this type of output. https://www.childsmath.ca/labs/R/prop_test.php 1-sample proportions test without continuity correction data: ? out of 60, null probability 0.6 X-squared = 1.1111, df = 1, p-value = ? alternative hypothesis: true p is not equal to 0.6 98 percent confidence interval: ? ? sample estimates: p ? ( .2938) ans: 2. The possible synergetic effect of insecticides and herbicides is a matter of concern to many environmentalists. It is feared that farmers who apply both herbicides and insecticides to a crop may enhance the toxicity of the insecticide beyond the desired level. An experiment is conducted with a particular insecticide and herbicide to determine the toxicity of three treatments: Treatment 1 : A concentration of .25 g per gram of soil insecticide with no herbicide . Treatment 2 : Same dosage of insecticide used in treatment 1 plus 100 g of herbicide per . gram of soil Treatment 3 : Same dosage of insecticide used in treatment 1 plus 400 g of herbicide per . gram of soil

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4 Several batches of fruit flies are exposed to each treatment, and the mortality percent is recorded as a measure of toxicity. The following data are obtained: Treatment 1: Treatment 2: Treatment 3: 40, 28, 31, 26, 33 38, 49, 56 68, 51, 45, 75 (a) Determine if the data indicate different toxicity levels among the three treatments using α œ Þ!& L  . ( Reject since 9.19 4.2565) ans: ! (b) Calculate the residuals corresponding to the data from Treatment 2. ( -9.67, 1.33, 8.33) ans: 3. It is believed that women in the postmenopausal phase of life suffer from calcium deficiency. This phenomenon is associated with the relatively high proportion of bone fractures for women in that age group. Is this calcium deficiency associated with an estrogen deficiency, a condition that occurs after menopause? To investigate this theory, Richelson, Wahner, Melton, and Riggs compared the bone mineral density in the three groups of women. The first group of 14 women had undergone oophorectomy during young adult womanhood and had lived for a period of 15 to 25 years with an estrogen deficiency. A second group of 15 women, identified as premenopausal, were approximately the same age (approximately 50 years) as the oophorectomy group except that the women had never suffered a period of estrogen deficiency. The third group of 16 women were postmenopausal and had suffered an estrogen deficiency for an average of 20 years. The mean and standard deviation for the three samples of lumbar spine bone-density measurements are recorded in the following table Group 1 ( Group 2 Group 3 Mean Mean Mean Standard Deviation St Oophorectomized) (Premenopausal) (Postmenopausal) : 0.93 : 1.21 : 0.92 : 0.04 andard Deviation Standard Deviation : 0.03 : 0.04 Source: L.S. Richelson, H.W. Wahner, L.J. Melton, III, and B.L. Riggs, "Relative Contributions of Aging and Estrogen Deficiency to Postmenopausal Bone Loss," New England Journal of Medicine 311(20) (1984), pp. 1273-75. Use Analysis of Variance with to test whether or not all three groups of women (a) α œ Þ!& have the same mean bone-density measurements. ( Reject since 298.13 3.2317) ans: L  ! (b) Use Bonferonni method with to determine which pairs of means are significantly α œ Þ!$ different. ( 1,2: Reject since 20.38 2.704; 1,3: Do not reject since ans: L  L ! ! . 2.704, 2,3: Reject since 21.83 2.704) ($*  L  !

5 4. Consider the data set summarized in the R output below. (a) Fill in the entries that have a question mark. (b) What hypothesis is tested in the ANOVA table, and what is the conclusion (using α œ Þ!"Ñ ? (c) Which pairs of means are significantly different at the 5% level? (d) What assumptions are required for the analysis? ( given on the page 6) ans: Df Sum Sq Mean Sq F value Pr(>F) exercise ? ? ? 0.00504 ** Residuals ? ? ? --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Pairwise comparisons using t tests with pooled SD data: systolic and exercise 0 1 2 1 0.250 - - 2 0.074 1.000 - 3 0.013 0.482 1.000 P value adjustment method: bonferroni exercise n mean sd 0 38 135.16 13.04 1 38 129.42 12.12 2 13 125.23 9.71 3 11 122.09 11.09

6 Answers for Suggested Problem #4: (a) Df Sum Sq Mean Sq F value Pr(>F) exercise 3 2001 667.1 4.547 0.00504 ** Residuals 96 14084 146.7 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Pairwise comparisons using t tests with pooled SD data: systolic and exercise 0 1 2 1 0.250 - - 2 0.074 1.000 - 3 0.013 0.482 1.000 P value adjustment method: bonferroni exercise n mean sd 0 38 135.16 13.04 1 38 129.42 12.12 2 13 125.23 9.71 3 11 122.09 11.09 (b) vs for at least one pair . Reject since L À œ œ œ À Á Ð3ß 4Ñ L ! " # $ % 3 4 ! . . . . . . L 1 Þ!!&  Þ!" . (c) Only 0 and 3. (d) The populations (or equivalently the residuals) must follow a normal distribution, and the population variances must be equal ( ). Also the 4 samples must be 5 5 5 5 " # $ # # # # % œ œ œ independent, and all other factors (other than ) affecting the variable EXERCISE SYSTOLIC must be kept constant.

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7 Suggested Problems #5: (Not to be handed in) 10.1 #5(reject since ),17(do not reject since ) L #&Þ)#  #Þ!!* L $Þ")#  "Þ#(  $Þ")# ! ! 10.2 #5,17 10.3 #9,17 Ch. 10 Review #5 11.1 #7,14( reject since ),15 ans: L ""Þ&&#  &Þ**" ! 11.2 #1,5,7,13 Ch. 11 Review #1 16. Given the five pairs of values, ÐBß CÑ B ! " C 2 4 5 9 8 5 7 4 (a) Construct a plot of residuals vs fitted values. (b) Estimate the expected value of corresponding to and give a 95% confidence C B œ #Þ& interval. (c) Predict the value of a single response corresponding to and give a 95% C B œ #Þ& prediction interval. (d) Construct the ANOVA table, and use it to test the hypothesis Test vs L À œ ! ! 3 L À Á ! œ Þ!& " 3 α using . (e) What proportion of the -variability is explained by the linear regression on ? C B (f) Calculate the sample correlation coefficient . < [ plot the pairs (.55814, 8.44186), (.32558, 7.67442), (-1.90698, 6.90698), (1.62791, ans: 5.37209), (-.60465, 4.60465); (4.335, 8.711); (1.170, 11.876); Do not Reject since L ! 4.3 10.13; .589; -.767]  17. Consider the R Output below. (a) Fill in the missing entries (the ones with a question mark). (b) What hypothesis is being tested, and what is the conclusion? (c) What assumptions are required for the validity of the chi-square analysis? ( given on p.8) ans: gender smoking F M All 0 25 ? 47 1 ? 19 37 2 7 9 16 All ? 50 ? Pearson's Chi-squared test data: smoking.status and gender X-squared = ?, df = ?, p-value = 0.7912

8 Answer for Suggested Problem #17 (a) gender smoking F M All 0 25 22 47 1 18 19 37 2 7 9 16 All 50 50 100 Pearson's Chi-squared test data: smoking.status and gender X-squared = 0.46852, df = 2, p-value = 0.7912 (b) Smoking status and gender are independent. Do not reject since .791 L À L  Þ!& ! ! (c) All of the expected values must be at least 5.